Cohen-Macaulayness of Vertex Splittable Monomial Ideals

1. Introduction

Inspired by the notion of vertex decomposable simplicial complexes, in [21] Moradi and Khosh-Ahang introduced the notion of vertex splittable monomial ideals. In this article, we characterize the Cohen-Macaulay vertex splittable monomial ideals (Theorem 1). Our result provides a simple, but very neat and effective, inductive strategy to classify Cohen-Macaulay monomial ideals. As a consequence, we recover the Cohen-Macaulay classifications of (vector-spread) strongly stable ideals, and (componentwise) polymatroidal ideals. Finally, a new characterization of bi-Cohen-Macaulay graphs is presented (Theorem 4).

2. Vertex Splittable Monomial Ideals

Let $S=K[x_1,\cdots ,x_n]$ be the polynomial ring over a field K, and let $I\subset S$ be a monomial ideal. We say that I is Cohen-Macaulay if $S/I$ is a Cohen-Macaulay ring. We denote by $\mathcal{G}\left(I\right)$ the unique minimal monomial generating set of I. We recall the following notion [21].

Definition 1.

The ideal I is called vertex splittable if it can be obtained by the following recursive procedure.

(i)

If u is a monomial and $I=\left(u\right)$, $I=0$ or $I=S$, then I is vertex splittable.

(ii)

If there exists a variable $x_i$ and vertex splittable ideals $I_1\subset S$ and $I_2\subset K[x_1,\cdots ,x_{i-1},$

$x_{i+1},\cdots ,x_n]$ such that $I=x_i{I}_1+I_2$, $I_2\subseteq I_1$ and $\mathcal{G}\left(I\right)$ is the union of $\mathcal{G}\left(x_i{I}_1\right)$ and $\mathcal{G}\left(I_2\right)$, then I is vertex splittable. In this case, we say that $I=x_i{I}_1+I_2$ is a vertex splitting of I and $x_i$ is a splitting vertex of I.

In contrast to arbitrary monomial ideals ([2], Page 236), the Cohen-Macaulayness of $S/I$, where I is vertex splittable, is independent from the field K. Indeed, the Krull dimension of $S/I$ does not depend on K. Furthermore, by [21], Theorem 2.4, vertex splittable ideals have linear quotients. Hence, $depthS/I$ is also independent from K [17], Corollary 8.2.2.

For recent applications of vertex splittings see the papers [4,5,6,22].

3. A Cohen-Macaulay Criterion

Let $\mathfrak{m}=(x_1,\cdots ,x_n)$ be the unique maximal homogeneous ideal of S.

Theorem 1.

Let $I\subset S$ be a vertex splittable monomial ideal such that $I\subseteq {\mathfrak{m}}^2$ and let $x_i$ be a splitting vertex of I. Then, the following conditions are equivalent.

(a)

I is Cohen-Macaulay.

(b)

$(I:x_i),(I,x_i)$ are Cohen-Macaulay and $depthS/(I:x_i)=depthS/(I,x_i)$.

Proof. 

We may assume $i=1$. Let $I=x_1{I}_1+I_2$ be the vertex splitting of I. Since $I=x_i{I}_1+I_2$ is a Betti splitting [21, Theorem 2.8], then [12, Corollary 2.2(b)] together with the Auslander-Buchsbaum formula, implies

$$depthS/I=min\{depthS/\left(x_1{I}_1\right),depthS/I_2,depthS/(x_1{I}_1\cap I_2)-1\}.$$

Notice that $depthS/x_1{I}_1=depthS/I_1$ and $x_1{I}_1\cap I_2=x_1(I_1\cap I_2)=x_1{I}_2$, because $I_2\subseteq I_1$ and $x_1$ does not divide any minimal monomial generator of $I_2$. Consequently, $depthS/(x_1{I}_1\cap I_2)=depthS/\left(x_1{I}_2\right)=depthS/I_2$, and so

$$depthS/I=min\{depthS/I_1,depthS/I_2-1\}.$$

(1)

We have the short exact sequence

$$0\to S/(I:x_1)\to S/I\to S/(I,x_1)\to 0.$$

Notice that $(I:x_1)=(x_1{I}_1+I_2):x_1=(x_1{I}_1:x_1)+(I_2:x_1)=I_1+I_2=I_1$, because $x_1$ does not divide any minimal monomial generator of $I_2$ and $I_2\subseteq I_1$. Since $I\subseteq {\mathfrak{m}}^2$, we have $x_1\notin I$. Thus $I_1\ne S$. Moreover, $(I,x_1)=(x_1{I}_1+I_2,x_1)=(I_2,x_1)$ and so we obtain the short exact sequence

$$0\to S/I_1\to S/I\to S/(I_2,x_1)\to 0.$$

Hence $dimS/I=max\{dimS/I_1,dimS/(I_2,x_1)\}$. Since $S/(I_2,x_1)\cong K[x_2,\cdots ,x_n]/I_2$ we obtain that $dimS/(I_2,x_1)=dimK[x_2,\cdots ,x_n]/I_2=dimS/I_2-1$. Hence,

$$dimS/I=max\{dimS/I_1,dimS/I_2-1\}.$$

(2)

(a)⇒(b) Suppose that I is Cohen-Macaulay. By equations (1) and (2) we have

$$dimS/I\ge dimS/I_1\ge depthS/I_1\ge depthS/I=dimS/I,$$

and

$$dimS/I\ge dimS/I_2-1\ge depthS/I_2-1\ge depthS/I=dimS/I.$$

Hence $S/I_1$, $S/I_2$ are Cohen-Macaulay and $depthS/I_1=depthS/I=depthS/I_2-1$.

(b)⇒(a) Conversely, assume that $S/I_1$ and $S/I_2$ are Cohen-Macaulay and that $depthS/I_1=depthS/I_2-1$. Then,

$$min\{depthS/I_1,depthS/I_2-1\}=depthS/I_1,$$

and

$$max\{dimS/I_1,dimS/I_2-1\}=dimS/I_1.$$

Equations (1) and (2) imply that $depthS/I=depthS/I_1=dimS/I_1=dimS/I$ and so $S/I$ is Cohen-Macaulay. □

The assumption $I\subseteq {\mathfrak{m}}^2$ stated in Theorem 1 can always be fulfilled. Indeed, if $\mathcal{G}\left(I\right)=\{x_{t+1},\cdots ,x_n,u_1,\cdots ,u_m\}$ where $u_1,\cdots ,u_m$ are monomials of degree $\ge 2$, then $S/I\cong K[x_1,\cdots ,x_t]/(u_1,\cdots ,u_m)$ and $(u_1,\cdots ,u_m)\subseteq {(x_1,\cdots ,x_t)}^2$.

Let $I\subset S$ be a graded Cohen-Macaulay ideal, and let $p=pd(S/I)$ be the projective dimension of $S/I$. The Cohen-Macaulay type of $S/I$ is defined as the integer $CM-type(S/I)={\beta }_p(S/I)$. It is well-known that $S/I$ is Gorenstein if and only $CM-type(S/I)=1$. We say that I is Gorenstein if $S/I$ is such.

By [18, Corollary 2.17] the graded Betti number ${\beta }_{p,p+regS/I}(S/I)$ is always non-zero. Here $regS/I$ is the Castelnuovo-Mumford regularity of $S/I$. We say that $S/I$ is level if and only if ${\beta }_p(S/I)={\beta }_{p,p+regS/I}(S/I)$. Following [8], we say that $S/I$ is pseudo-Gorenstein if and only if ${\beta }_{p,p+regS/I}(S/I)=1$. Hence $S/I$ is Gorenstein if and only if it is both level and pseudo-Gorenstein.

Finally, for a finitely generated graded S-module M, we denote by $H_{\mathfrak{m}}^i\left(M\right)$ the ith local cohomology module of M with support on $\mathfrak{m}$.

Corollary 1.

Let $I\subset S$ be a vertex splittable Cohen-Macaulay ideal such that $I\subseteq {\mathfrak{m}}^2$ and let $I=x_i{I}_1+I_2$ be a vertex splitting of I. Then, the following statements hold.

(a)

$CM-type(S/I)=CM-type(S/I_1)+CM-type(S/I_2)$.

(b)

$S/I$ is Gorenstein if and only if I is a principal ideal.

(c)

$S/I$ is level if and only if $S/I_1$ and $S/I_2$ are level and $regS/I_1+1=regS/I_2$.

(d)

$S/I$ is pseudo-Gorenstein if and only if

(i)

either $S/I_1$ is pseudo-Gorenstein and $regS/I_1+1>regS/I_2$, or

(ii)

$S/I_2$ is pseudo-Gorenstein and $regS/I_1+1<regS/I_2$.

(e)

$H_{\mathfrak{m}}^{dimS/I}(S/I)/H_{\mathfrak{m}}^{dimS/I}(S/(I:x_i))\cong H_{\mathfrak{m}}^{dimS/I}(S/(I,x_i))$.

Proof. 

We may assume that $x_i=x_1$. Since $S/I$ is Cohen-Macaulay, Theorem 1 guarantees that $S/I_1,S/I_2$ are Cohen-Macaulay and $depthS/I_1=depthS/I_2-1$. Hence $pdS/I_1=pdS/I_2+1$. Let $p=pdS/I_1$. In particular, we have $p=pdS/I$. Now, by [21, Remark 2.10] we have for all j

$${\beta }_{p,p+j}(S/I)={\beta }_{p,p+j-1}(S/I_1)+{\beta }_{p,p+j}(S/I_2)+{\beta }_{p-1,p-1+j}(S/I_2).$$

Since $pdS/I_2=p-1$, the above formula simplifies to

$${\beta }_{p,p+j}(S/I)={\beta }_{p,p+j-1}(S/I_1)+{\beta }_{p-1,p-1+j}(S/I_2).$$

(3)

From this formula we deduce that

$$regS/I=max\{regS/I_1+1regS/I_2\}.$$

(a) The assertion follows immediately by equation (3).

(b) In the proof of Theorem 1 we noted that $I_1\ne 0,S$. Thus $CM-type(S/I_1)\ge 1$. By (a), it follows that I is Gorenstein if and only if $CM-type(S/I_1)=1$, that is $I_1$ is Gorenstein, and $CM-type(S/I_2)=0$, that is $I_2=0$. But, if $I_2=0$ then $depthS/I_2=depthS=n$, and so by formula (1) and Theorem 1(b) we obtain that $depthS/I=depthS/I_1=depthS/I_2-1=n-1$. Thus $pdI=0$, and this is possible if and only if I is a principal ideal.

(c) Assume that $S/I$ is level. Then ${\beta }_{p,p+j}(S/I)\ne 0$ only for $j=regS/I$. Since $regS/I=max\{regS/I_1+1,regS/I_2\}$ and ${\beta }_{p,p+regS/I_1}(S/I_1),{\beta }_{p-1,p-1+regS/I_2}(S/I_2)$ are both non-zero, we deduce from formula (3) that $regS/I_1+1=regS/I_2$ and that $S/I_1$, $S/I_2$ must be level. Conversely, if $regS/I_1+1=regS/I_2$ and $S/I_1$, $S/I_2$ are level, we deduce from formula (3) that $S/I$ is level.

(d) Assume that $S/I$ is pseudo-Gorenstein. Then ${\beta }_{p,p+regS/I}(S/I)=1$. Since $regS/I=max\{regS/I_1+1,regS/I_2\}$ and ${\beta }_{p,p+regS/I_1}(S/I_1),{\beta }_{p-1,p-1+regS/I_2}(S/I_2)$ are both non-zero, we deduce from formula (3) that either $S/I_1$ is pseudo-Gorenstein and $regS/I_1+1>regS/I_2$, or $S/I_2$ is pseudo-Gorenstein and $regS/I_1+1<regS/I_2$. The converse can be proved in a similar way.

(e) Since $S/I$ is Cohen-Macaulay, Theorem 1 implies that $S/(I:x_i)$ and $S/(I,x_i)$ are Cohen-Macaulay and $dimS/(I:x_i)=dimS/(I,x_i)=dimS/I$. As shown in the proof of Theorem 1 we have the short exact sequence

$$0\to S/(I:x_i)\to S/I\to S/(I,x_i)\to 0.$$

This sequence induces the long exact sequence of local cohomology modules:

$$\cdots \to H_{\mathfrak{m}}^{i-1}(S/(I,x_i))\to H_{\mathfrak{m}}^i(S/(I:x_i))\to H_{\mathfrak{m}}^i(S/I)\to H_{\mathfrak{m}}^i(S/(I,x_i))\to \cdots .$$

Let M be a finitely generated Cohen-Macaulay S-module. By Grothendieck’s vanishing theorem [2, Theorem 5.3.7], $H_{\mathfrak{m}}^i\left(M\right)\ne 0$ if and only if $i=depthM=dimM$. Thus, the above exact sequence simplifies to

$$0\to H_{\mathfrak{m}}^{dimS/I}(S/(I:x_i))\to H_{\mathfrak{m}}^{dimS/I}(S/I)\to H_{\mathfrak{m}}^{dimS/I}(S/(I,x_i))\to 0,$$

and the assertion follows. □

It is clear that any ideal $I\subset S$ generated by a subset of the variables of S is Gorenstein and vertex splittable. Hence, Corollary 1 implies immediately

Corollary 2.

The only Gorenstein vertex splittable ideals of S are the principal monomial ideals and the ideals generated by a subset of the variables.

4. Families of Cohen-Macaulay Vertex Splittable Ideals

In this section, by using Theorem 1 we recover in a simple and very effective manner several Cohen-Macaulay classification of families of monomial ideals. We use the fact that if $I=x_i{I}_1+I_2$ is a vertex splitting, then $I_1,I_2$ are vertex splittable ideals that, in good cases, belong again to a given family of vertex splittable monomial ideals, and to which one may apply inductive arguments.

4.1. (Vector-Spread) Strongly Stable Ideals

Strongly stable monomial ideals are fundamental in Commutative Algebra, because if K has characteristic zero, then they appear as generic initial ideals [17]. In [9], the concept of strongly stable ideal has been generalized to that of $\mathbf{t}$-spread strongly stable ideal.

Let $d\ge 2$, $\mathbf{t}=(t_1,\cdots ,t_{d-1})\in {\mathbb{Z}}_{\ge 0}^{d-1}$ be a $(d-1)$tuple, and let $u=x_{i_1}\cdots x_{i_{\ell }}\in S$ be a monomial, with $1\le i_1\le \cdots \le i_{\ell }\le n$ and $\ell \le d$. We say that u is $\mathbf{t}$-spread if

$$i_{j+1}-i_j\ge t_j\mathrm{for} \mathrm{all}j=1,\cdots ,\ell -1.$$

A monomial ideal $I\subset S$ is called $\mathbf{t}$-spread if $\mathcal{G}\left(I\right)$ consists of $\mathbf{t}$-spread monomials. A $\mathbf{t}$-spread ideal $I\subset S$ is called $\mathbf{t}$-spread strongly stable if for all $\mathbf{t}$-spread monomials $u\in I$ and all $i<j$ such that $x_j$ divides u and $x_i(u/x_j)$ is $\mathbf{t}$-spread, then $x_i(u/x_j)\in I$. For $\mathbf{t}=(0,\cdots ,0)$ and $\mathbf{t}=(1,\cdots ,1)$, we obtain the strongly stable and the squarefree strongly stable ideals. In [3] we classified the Cohen-Macaulay $\mathbf{t}$-spread strongly stable ideals. We recover this result using Theorem 1.

Corollary 3.

Let $I\subset S$ be a $\mathbf{t}$-spread strongly stable ideal such that $I\subseteq {\mathfrak{m}}^2$. Then I is Cohen-Macaulay if and only if there exists $\ell \le d$ such that

$$x_{n-(t_1+t_2\cdots +t_{\ell -1})}{x}_{n-(t_2+t_3\cdots +t_{\ell -1})}\cdots x_{n-t_{\ell -1}}{x}_n\in \mathcal{G}\left(I\right).$$

Proof. 

Firstly, we prove that I is vertex splittable. If I is a principal ideal, then it is vertex splittable. Suppose now that $\left|\mathcal{G}\right(I\left)\right|>1$. We can write $I=x_1{I}_1+I_2$, where $\mathcal{G}\left(I_1\right)=\{u/x_1:u\in \mathcal{G}\left(I\right),x_{1}dividesu\}$ and $\mathcal{G}\left(I_2\right)=\mathcal{G}\left(I\right)\setminus \mathcal{G}\left(x_1{I}_1\right)$. It is immediate to see that $I_1\subset S$ is $(t_2,\cdots ,t_{d-1})$-spread strongly stable and that $I_2$ is a $\mathbf{t}$-spread strongly stable ideal of $K[x_2,\cdots ,x_n]$. By induction on n and on the highest degree of a generator of I, we have that $I_1$ and $I_2$ are vertex splittable. Hence, so is I.

We may suppose that $x_n$ divides some minimal generator of I. Otherwise, we can consider I as a monomial ideal of a smaller polynomial ring. If I is principal, then we have $I=\left(u\right)=(x_1{x}_{1+t_1}\cdots x_{1+t_1+\cdots +t_{\ell -1}})$, with $n=1+t_1+\cdots +t_{\ell -1}$, and $\ell \le d$. Otherwise, if I is not principal, then $pdS/I>1$ and we can write $I=x_1{I}_1+I_2$ as above. By Theorem 1, $I_2$ is Cohen-Macaulay and $pdS/I_2=pdS/I+1>1$. Thus $I_2\ne 0$. Hence, by induction there exists $\ell \le d$ such that

$$x_{n-(t_1+\cdots +t_{\ell -1})}\cdots x_{n-t_{\ell -1}}{x}_n\in \mathcal{G}\left(I_2\right).$$

Since $\mathcal{G}\left(I_2\right)\subset \mathcal{G}\left(I\right)$, the assertion follows. □

4.2. Componentwise Polymatroidal Ideals

For a monomial ideal $I\subset S$ and an integer $j\ge 0$, we denote by $I_{\langle j\rangle }$, the jth component of I, that is, the monomial ideal of S generated by all monomials of degree j belonging to I.

Let $I\subset S$ be a monomial generated in a single degree. We say that I is polymatroidal if the set of the exponent vectors of the minimal monomial generators of I is the set of bases of a discrete polymatroid [15]. An arbitrary monomial ideal I is called componentwise polymatroidal if the component $I_{\langle j\rangle }$ is polymatroidal for all j.

Polymatroidal ideals are characterized by the exchange property [15]. For a monomial $u\in S$, let

$${deg}_{x_i}\left(u\right)=max\{j:x_i^{j}dividesu\}.$$

A monomial ideal $I\subset S$ generated in a single degree is polymatroidal if and only if for all $u,v\in \mathcal{G}\left(I\right)$ and all i such that ${deg}_{x_i}\left(u\right)>{deg}_{x_i}\left(v\right)$, there exists j with ${deg}_{x_j}\left(u\right)<{deg}_{x_j}\left(v\right)$ such that $x_j(u/x_i)\in \mathcal{G}\left(I\right)$.

A longstanding conjecture of Bandari and Herzog predicted that componentwise polymatroidal ideals have linear quotients [1]. This conjecture has been solved recently in [10]. Inspecting the proof of that theorem, we notice the next result. For the polymatroidal case see also [20].

Proposition 1.

Componentwise polymatroidal ideals are vertex splittable.

Proof. 

We may assume that all variables $x_i$ divide some minimal monomial generator of I. It is noted in the proof of [10] that for any variable $x_i$ which divides a minimal monomial generator of minimal degree of I, we can write $I=x_i{I}_1+I_2$, where $\mathcal{G}\left(x_i{I}_1\right)=\{u\in \mathcal{G}\left(I\right):x_{i}dividesu\}$, $\mathcal{G}\left(I_2\right)=\mathcal{G}\left(I\right)\setminus \mathcal{G}\left(x_i{I}_1\right)$ and the following properties are satisfied:

(a)

$I_2\subseteq I_1$ as monomial ideals of S.

(b)

$x_i{I}_1$ is a componentwise polymatroidal ideal of S.

(c)

$I_2$ is a componentwise polymatroidal ideal of $K[x_1,\cdots ,x_{i-1},x_{i+1},\cdots ,x_n]$.

By induction on n and on the highest degree of a generator of I, it follows that both $I_1$ and $I_2$ are vertex splittable. Hence, so is I. □

Combining the vertex splitting presented in the above proof with the facts (b) and (c) and with Theorem 1, we obtain the following immediate corollary.

Corollary 4.

Let $I\subset S$ be a componentwise polymatroidal ideal and let $x_i$ be any variable dividing some minimal monomial generator of least degree of I. Suppose that $I\subseteq {\mathfrak{m}}^2$. Then, the following conditions are equivalent.

(a)

I is Cohen-Macaulay.

(b)

$(I:x_i),(I,x_i)$ are Cohen-Macaulay componentwise polymatroidal ideals and $depthS/(I:x_i)=depthS/(I,x_i)$.

Corollary 5.

[11]Let $I\subset S$ be a polymatroidal ideal generated in degree $d\ge 2$ and let $x_i$ be a variable dividing some monomial of $\mathcal{G}\left(I\right)$. Then $(I:x_i)$ is polymatroidal. If, in addition, I is Cohen-Macaulay, then $(I:x_i)$ is also Cohen-Macaulay.

At the moment, to classify all Cohen-Macaulay componentwise polymatroidal ideals seems to be an hopeless task. Indeed, let J be any componentwise polymatroidal ideal. Let ℓ be the highest degree of a minimal monomial generator of J, and let $d>\ell$ be any integer. It is easy to see that $I=J+{\mathfrak{m}}^d$ is componentwise polymatroidal. Since $dimS/I=0$, then $S/I$ is automatically Cohen-Macaulay.

For another example, consider the ideal $I=(x_1^2,x_1{x}_3,x_3^2,x_1{x}_2{x}_4,x_2{x}_3{x}_4,x_2^2{x}_4^2)$ of $S=K[x_1,x_2,x_3,x_4]$, see [10]. One can easily check that I is a Cohen-Macaulay componentwise polymatroidal ideal. In this case, $dimS/I>0$.

Nonetheless, if I is generated in a single degree, that is, if I is actually polymatroidal, then Herzog and Hibi [16] showed that I is Cohen-Macaulay if and only if: (i) I is a principal ideal, (ii) I is a squarefree Veronese ideal$I_{n,d}$, that is, it is generated by all squarefree monomials of S of a given degree $d\le n$, or (iii) I is a Veronese ideal, that is, $I={\mathfrak{m}}^d$ for some integer $d\ge 1$.

The proof presented by Herzog and Hibi is based on the computation of $\sqrt{I}$. We now present a different proof based on Theorem 1.

Corollary 6.

A polymatroidal $I\subset S$ is Cohen-Macaulay if and only if I is

(a)

a principal ideal,

(b)

a Veronese ideal, or

(c)

a squarefree Veronese ideal.

For the proof, we need the following well-known identities. For the convenience of the reader, we provide a proof that uses the vertex splittings technique.

Lemma 1.

Let $n,d\ge 1$ be positive integers. Then

$$pdS/{\mathfrak{m}}^d=n,\mathrm{and}pdS/I_{n,d}=n+1-d.$$

Proof. 

Since $dimS/{\mathfrak{m}}^d=0$ we have $depthS/{\mathfrak{m}}^d=0$ and $pdS/{\mathfrak{m}}^d=n$. If $d=1$, then $I_{n,1}=\mathfrak{m}$ and $pdS/I_{n,1}=n$. If $1<d\le n$, we notice that $I_{n,d}=x_n{I}_{n-1,d-1}+I_{n-1,d}$ is a vertex splitting. By formula (1) and induction on n and d,

$$\begin{array}{cc}\hfill pdS/I_{n,d}& =min\{pdS/I_{n-1,d-1},pdS/I_{n-1,d}+1\}\hfill \\ \hfill & =min\{n-1+1-(d-1),n-1+1-d+1\}\hfill \\ \hfill & =n+d-1,\hfill \end{array}$$

as wanted □

We are now ready for the proof of Corollary 6.

Proof of Corollary 6

Let I be a polymatroidal ideal. If I is principal, then it is Cohen-Macaulay. Now, let $\left|\mathcal{G}\right(I\left)\right|>1$. If $I=\mathfrak{m}$, then I is Cohen-Macaulay. Thus, we assume that I is generated in degree $d\ge 2$, and that all variables $x_i$ divide some minimal monomial generator of I. By Proposition 1 and Corollaries 5, 5, we have a vertex splitting $I=x_i{I}_1+I_2$ for each variable $x_i$, and $I_1$ and $I_2$ are Cohen-Macaulay polymatroidal ideals with $depthS/I_1=depthS/I_2-1$. Thus $pdS/I_1=pdS/I_2+1$. We may assume that $x_i=x_n$.

By induction on n and on d, it follows that $I_1$ is either a principal ideal, a Veronese ideal, or a squarefree Veronese ideal, and the same possibilities occur for $I_2$. We distinguish the various possibilities.

Case 1. Let $I_2$ be a principal ideal, then $pdS/I_2=1$. Thus $pdS/I_1=2$.

Under this assumption, $I_1$ cannot be principal because $pdS/I_1=2\ge 1$.

Assume that $I_1$ is a Veronese ideal in m variables. Since all variables of S divide some monomial of $\mathcal{G}\left(I\right)$ and $I_2\subset I_1$, then $I_1={\mathfrak{m}}^d$ or $I_1={\mathfrak{n}}^d$, where $\mathfrak{m}=(x_1,\cdots ,x_n)$ and $\mathfrak{n}=(x_1,\cdots ,x_{n-1})$. Thus $m=n$ or $m=n-1$. Lemma 1 implies $m=pdS/I_1=2$. So, $n=2$ or $n=3$.

If $n=2$, then $I=x_2{(x_1,x_2)}^{d-1}+\left(x_1^d\right)={(x_1,x_2)}^d$ which is Cohen-Macaulay and Veronese, or $I=x_2\left(x_1^{d-1}\right)+\left(x_1^d\right)=x_1^{d-1}(x_1,x_2)$ which is not Cohen-Macaulay.

Otherwise, if $n=3$ then $I=x_3{(x_1,x_2,x_3)}^{d-1}+\left(u\right)$ or else $I=x_3{(x_1,x_2)}^{d-1}+\left(u\right)$ where $u\in K[x_1,x_2]$ is a monomial of degree d. If $d=2$, then one easily sees that only in the second case and for $u=x_1{x}_2$ we have that I a Cohen-Macaulay polymatroidal ideal, which is the squarefree Veronese $I_{3,2}$. Otherwise, suppose $d\ge 3$. We may assume that $x_1^2$ divides u. In the first case, $(I:x_1)=x_3{(x_1,x_2,x_3)}^{d-2}+(u/x_1)$ is not principal, neither Veronese, neither squarefree Veronese. Thus, by induction $(I:x_1)$ is not a Cohen-Macaulay polymatroidal ideal, and by Corollary 5 we deduce that I is also not Cohen-Macaulay. Similarly, in the second case, we see that $(I:x_1)=x_3{(x_1,x_2)}^{d-1}+(u/x_1)$, and thus also I, is not a Cohen-Macaulay polymatroidal ideal.

Assume now that $I_1$ is a squarefree Veronese ideal in m variables. Then as argued in the case 1.2 we have $m=n$ or $m=n-1$. Lemma 1 gives $pdS/I_1=m+1-(d-1)=2$. Thus $d=m$. Hence $d=n$ or $d=n-1$. So $I=x_n{I}_{n,n-1}+\left(u\right)$ or $I=x_n{I}_{n-1,n-2}+\left(u\right)$ where $u\in K[x_1,\cdots ,x_{n-1}]$ is a monomial of degree $d=n$ in the first case or $d=n-1$ in the second case. In the first case there is i such that $x_i^2$ divides u. Say $i=1$. Then $(I:x_1)=x_n(I_{n,n-1}:x_1)+(u/x_1)$ is not a principal ideal, neither a Veronese ideal, neither a squarefree Veronese. Therefore, by induction $(I:x_1)$ we see that is not a Cohen-Macaulay polymatroidal ideal, and by Corollary 5I is also not a Cohen-Macaulay polymatroidal ideal. Similarly, in the second case, if $u=x_1\cdots x_{n-1}$ then $I=I_{n,n-1}$ is a Cohen-Macaulay squarefree Veronese ideal. Otherwise $x_i^2$, say with $i=1$, divides u and then, arguing as before, we see that I is not a Cohen-Macaulay polymatroidal ideal.

Case 2. Let $I_2$ be a Veronese ideal in m variables, then $pdS/I_2=m\le n-1$.

Under this assumption, $I_1$ cannot be principal because $pdS/I_1=m+1>1$.

Assume that $I_1$ is a Veronese ideal in ℓ variables. Then $\ell =n$ or $\ell =n-1$. Lemma 1 implies that $\ell =pdS/I_1=pdS/I_2+1=m+1$. Thus $\ell =n$ and $m=n-1$ or $\ell =n-1$ and $m=n-2$. In the first case $I=x_n{\mathfrak{m}}^{d-1}+{(x_1,\cdots ,x_{n-1})}^d={\mathfrak{m}}^d$ is a Cohen-Macaulay Veronese ideal. In the second case, up to relabeling we can write $I=x_n{(x_1,\cdots ,x_{n-1})}^{d-1}+{(x_1,\cdots ,x_{n-2})}^d$. But this ideal is not polymatroidal. Otherwise, by the exchange property applied to $u=x_n{x}_{n-1}^{d-1}$ and $v=x_{n-2}^d$, we should have $x_{n-2}{x}_{n-1}^{d-1}\in I$, which is not the case.

Assume now that $I_1$ is a squarefree Veronese ideal in ℓ variables.Then $\ell =n$ or $\ell =n-1$. Lemma 1 implies that $pdS/I_1=\ell +1-(d-1)=\ell +2-d=pdS/I_2+1=m+1$. Thus, either $m=n+1-d$ or $m=n-d$. Up to relabeling, we have either $I=x_n{I}_{n,d-1}+{(x_1,\cdots ,x_{n+1-d})}^d$ or $I=x_n{I}_{n-1,d-1}+{(x_1,\cdots ,x_{n-d})}^d$. If $d=2$, then these ideals become either $I={(x_1,\cdots ,x_n)}^2$ which is a Cohen-Macaulay Veronese ideal, or $I=x_n(x_1,\cdots ,x_{n-1})+{(x_1,\cdots ,x_{n-2})}^2$ which is not polymatroidal because the exchange property does not hold for $u=x_n{x}_{n-1}$ and $v=x_{n-2}^2$ since $x_{n-1}{x}_{n-2}\notin I$. If $d\ge 3$, then the above ideals are not polymatroidal. In the first case the exchange property does not hold for $u=x_{n+1-d}^d\in I_2$ and $v=(x_{n+2-d}\cdots x_n)x_n\in x_n{I}_1$, otherwise $x_j{x}_{n+1-d}^{d-1}\in I$ for some $n+2-d\le j\le n$, which is not the case.

Case 3. Let $I_2$ be a squarefree Veronese in m variables, $m\le n-1$. Then Lemma 1 implies $pdS/I_2=m+1-d$ with $d\le m$. Hence $pdS/I_1=m+2-d$.

In such a case the ideal $I_1$ cannot be principal because $pdS/I_1=m+2-d>1$.

Assume that $I_1$ is a Veronese ideal in ℓ variables. Then $\ell =n$ or $\ell =n-1$. Lemma 1 implies that $\ell =pdS/I_1=m+2-d$. Hence either $d=m-n+2$ or $d=m-n+3$. Since $m\le n-1$, either $d\le 1$ or $d\le 2$. Only the case $d=2$ is possible. If $d=2$, then $\ell =m=n-1$ and we have $I=x_n(x_1,\cdots ,x_{n-1})+{(x_1,\cdots ,x_{n-1})}^2$. This ideal is not Cohen-Macaulay, otherwise it would be height-unmixed. Indeed $(I:x_n)=(x_1,\cdots ,x_{n-1})$ and $(I:x_{n-1})=(x_1,\cdots ,x_n)$ are two associated primes of I having different height.

Finally, assume that $I_1$ is a squarefree Veronese ideal in ℓ variables.Then $\ell =n$ or $\ell =n-1$. Lemma 1 implies that $pdS/I_1=\ell +1-(d-1)=\ell +2-d=m+2-d$. Thus $\ell =m$ and so either $\ell =m=n$ or $\ell =m=n-1$. The first case is impossible because $m\le n-1$. In the second case, we have $I=x_n{I}_{n-1,d-1}+I_{n-1,d}=I_{n,d}$ which is a Cohen-Macaulay squarefree Veronese ideal. □

4.3. Bi-Cohen-Macaulay Graphs

Let $I\subset S$ be a squarefree monomial ideal. Then I may be seen as the Stanley-Reisner ideal of a unique simplicial complex on the vertex set $\{1,\cdots ,n\}$. Attached to I is the Alexander dual $I^{\vee }$, which is again a squarefree monomial ideal. We say that I is bi-Cohen-Macaulay if both I and $I^{\vee }$ are Cohen-Macaulay. By the Eagon-Reiner criterion [17, Theorem 8.1.9] I has a linear resolution if and only if $I^{\vee }$ is Cohen-Macaulay. Hence, I is bi-Cohen-Macaulay if and only if it is Cohen-Macaulay with linear resolution.

Let G be a finite simple graph on the vertex set $V\left(G\right)=\{1,\cdots ,n\}$ with edge set $E\left(G\right)$. The edge ideal$I\left(G\right)$ of G is the squarefree monomial ideal of S generated by the monomials $x_i{x}_j$ with $\{i,j\}\in E\left(G\right)$[23]. The Alexander dual of $I\left(G\right)$ is the squarefree monomial ideal of S generated by the squarefree monomial $x_{i_1}\cdots x_{i_t}$ such that $\{i_1,\dots ,i_t\}$ is a minimal vertex cover of G[23]. Such an ideal is denoted by $J\left(G\right)$ and, since its definition, it is often called the cover ideal of G.

We say that G is a bi-Cohen-Macaulay graph if $I\left(G\right)$ is bi-Cohen-Macaulay.

The edge ideals with linear resolution have been classified by Fröberg [13]. A graph G is complete if every $\{i,j\}$ with $i,j\in V\left(G\right)$, $i\ne j$, is an edge of G. The open neighbourhood of $i\in V\left(G\right)$ is the set

$$N_G\left(i\right)=\{j\in V\left(G\right):\{i,j\}\in E\left(G\right)\}.$$

A graph G is called chordal if it has no induced cycles of length bigger than three. A perfect elimination order of G is an ordering $v_1,\cdots ,v_n$ of its vertex set $V\left(G\right)$ such that $N_{G_i}\left(v_i\right)$ induces a complete subgraph on $G_i$, where $G_i$ is the induced subgraph of G on the vertex set $\{i,i+1,\cdots ,n\}$. Hereafter, if $1,2,\cdots ,n$ is a perfect elimination order of G, we highlight it by $x_1>x_2>\cdots >x_n$.

Theorem 2. (Dirac, [7]).A finite simple graph G is chordal if and only if G admits a perfect elimination order.

The complementary graph $G^c$ of G is the graph with vertex set $V\left(G^c\right)=V\left(G\right)$ and where $\{i,j\}$ is an edge of $G^c$ if and only if $\{i,j\}\notin E\left(G\right)$. A graph G is called cochordal if and only if $G^c$ is chordal.

Theorem 3. (Fröberg, [13]).Let G be a finite simple graph. Then, $I\left(G\right)$ has a linear resolution if and only if G is cochordal.

We quote the next fundamental result which was proved by Moradi and Khosh-Ahang [21, Theorem 3.6, Corollary 3.8].

Proposition 2.

Let G be a finite simple graph. Then $I\left(G\right)$ has linear resolution if and only if $I\left(G\right)$ is vertex splittable. Furthermore, if $x_1>\cdots >x_n$ is a perfect elimination order of $G^c$, then

$$I\left(G\right)=x_1(x_j:j\in N_G\left(1\right))+I(G\setminus \left\{1\right\})$$

is a vertex splitting of $I\left(G\right)$.

Combining the above result with Theorem 1 we obtain the next characterization of the bi-Cohen-Macaulay graphs.

Theorem 4.

For a finite simple graph G, the following conditions are equivalent.

(a)

G is a bi-Cohen-Macaulay graph.

(b)

$G^c$ is a chordal graph with perfect elimination order $x_1>\cdots >x_n$ and

$$|N_G\left(i\right)\cap \{i,\cdots ,n\}|=|N_G\left(j\right)\cap \{j,\cdots ,n\}|+(j-i),$$

for all $1\le i\le j\le n$ such that $|N_G\left(i\right)\cap \{i,\cdots ,n\}|,|N_G\left(j\right)\cap \{j,\cdots ,n\}|>0$.

In particular, if any of the equivalent conditions hold, then

$$pdS/I\left(G\right)=|N_G\left(i\right)\cap \{i,\cdots ,n\}|+(i-1),$$

for any $1\le i\le n$ such that $|N_G\left(i\right)\cap \{i,\cdots ,n\}|>0$.

Proof. 

We proceed by induction on $n=\left|V\right(G\left)\right|$. By Theorem 3, G must be cochordal. Fix $x_1>\cdots >x_n$ a perfect elimination order of $G^c$. By Proposition 2, $I\left(G\right)=x_1(x_j:j\in N_G\left(1\right))+I(G\setminus \left\{1\right\})$ is a vertex splitting. Applying Theorem 1, $I\left(G\right)$ is Cohen-Macaulay if and only if $J=(x_j:j\in N_G\left(1\right))$ and $I(G\setminus \{1\left\}\right)$ are Cohen-Macaulay and $pdS/I\left(G\right)=pdS/J=pdS/I(G\setminus \{1\left\}\right)+1$. J is Cohen-Macaulay because it is an ideal generated by variables and $pdS/J=|N_G\left(1\right)|$. Notice that $x_2>\cdots >x_n$ is a perfect elimination order of ${(G\setminus \left\{1\right\})}^c$.

If $I(G\setminus \{1\left\}\right)=0$, then $pdS/I\left(G\right)=pdS/J=pdS/I(G\setminus \{1\left\}\right)+1=1$ and $I\left(G\right)$ is principal, say $I\left(G\right)=\left(x_1{x}_2\right)$. In this case the thesis holds.

Suppose now $I(G\setminus \{1\left\}\right)\ne 0$. Then, by induction on n, $I(G\setminus \{1\left\}\right)$ is Cohen-Macaulay if and only if

$$|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|=|N_{G\setminus \left\{1\right\}}\left(j\right)\cap \{j,\cdots ,n\}|+(j-i),$$

(4)

for all $2\le i\le j\le n$ such that $|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|,|N_{G\setminus \left\{1\right\}}\left(j\right)\cap \{j,\cdots ,n\}|>0$ and moreover

$$pdS/I(G\setminus \left\{1\right\})=|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|+(i-2),$$

(5)

for any $2\le i\le n$ such that $|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|>0$.

Notice that $N_G\left(i\right)\cap \{i,\cdots ,n\}=N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}$ for all $2\le i\le n$. Thus, combining (4) and (5) with the equality $pdS/I\left(G\right)=pdS/J=pdS/I(G\setminus \{1\left\}\right)+1$, we see that $I\left(G\right)$ is Cohen-Macaulay if and only if

$$\begin{array}{cc}\hfill |N_G\left(1\right)|& =|N_G\left(1\right)\cap \{1,\cdots ,n\}|=|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|+(i-2)+1\hfill \\ \hfill & =|N_G\left(i\right)\cap \{i,\cdots ,n\}|+(i-1),\hfill \end{array}$$

for all $2\le i\le n$ such that $|N_{G\setminus \left\{1\right\}}\left(i\right)\cap \{i,\cdots ,n\}|>0$.

Thus, we deduce that $|N_G\left(i\right)\cap \{i,\cdots ,n\}|=|N_G\left(j\right)\cap \{j,\cdots ,n\}|+(j-i)$ for all $1\le i\le j\le n$ such that $|N_G\left(i\right)\cap \{i,\cdots ,n\}|,|N_G\left(j\right)\cap \{j,\cdots ,n\}|>0$, as desired. The inductive proof is complete. □

Notice that in the above characterization, the field K plays no role. In other words, the bi-Cohen-Macaulay property of edge ideals does not depend on the field K. This also follows from the work of Herzog and Rahimi [19, Corollary 1.2 (d)] where other classifications of the bi-Cohen-Macaulay graphs are given.

Example 1.

Consider the graph G on five vertices and its complementary graph $G^c$ depicted below in Figure 1.

Notice that $x_1>x_2>x_3>x_4>x_5$ is a perfect elimination order of $G^c$, so that $G^c$ is chordal (Theorem 2). We have $|N_G\left(i\right)\cap \{i,\cdots ,5\}|>0$ only for $i=1,2,3$. It is easy to see that condition (b) of Theorem 4 is verified. Hence G is bi-Cohen-Macaulay, as one can also verify by using Macaulay2 [14].

Example 2.

Consider the graph H and its complementary graph $H^c$ depicted below in Figure 2.

As before $x_1>x_2>x_3>x_4>x_5$ is a perfect elimination order of $H^c$, and $H^c$ is chordal. We have $|N_H\left(i\right)\cap \{i,\cdots ,5\}|>0$ only for $i=1,2,3$. However condition (b) of Theorem 4 is not verified. Indeed,

$$|N_H\left(1\right)\cap \{1,\cdots ,5\}|=|N_H\left(2\right)\cap \{2,\cdots ,5\}|=|\{4,5\}|=2,$$

but $|N_H\left(1\right)\cap \{1,\cdots ,5\}|\ne |N_H\left(2\right)\cap \{2,\cdots ,5\}|+1$. Hence H is not bi-Cohen-Macaulay. We can also verify this by using Macaulay2 [14]. Indeed $J\left(H\right)$, the cover ideal of H, is not Cohen-Macaulay.